Does $g(f(x))$ imply $g(x)$? If $g$ is a function of $f(x)$ does this imply that $g$ is a function of $x$?
If yes, am I allowed to write the chain rule as:
$$\frac{{d[g(f[x])]}}{{dx}} = \frac{{d[g(x)]}}{{d[f(x)]}} \cdot \frac{{d[f(x)]}}{{dx}}
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Thanks in advance
 A: It's perfectly fine to write the chain rule as
$$
\frac{\mathrm d[g(f(x))]}{\mathrm dx}=\frac{\mathrm d[g(f(x))]}{\mathrm d[f(x)]}\cdot\frac{\mathrm d[f(x)]}{\mathrm dx}.
$$
The first term just means "differentiate $g(f(x))$ with respect to $f(x)$", i.e. pretend $f(x)$ is the independent variable, then fix your error by multiplying with how $f(x)$ changes with $x$, i.e. $\mathrm d[f(x)]/\mathrm dx$. That's exactly how the chain rule works.
I think you are confusing $g$ with $g\circ f$. Those are two different maps. This confusion may arise from sloppy notation, like the notation often found in physics, where $g$ and $g(y)$ are used interchangably for a quantity $g$ that depends on another quantity $y$. A good notation get's rid of this confusion by focusing on the maps and their respective domains and ranges instead of quantities and what they depend on.
In this case the maps are $f:X\to Y$, $g:Y\to Z$ and $g\circ f:X\to Z$, where $X$, $Y$ and $Z$ are sets. So $f$ takes an element of the set $X$ to the set $Y$ which might then be mapped to an element in $Z$ by applying $g$. Using $f'$ and $g'$ for the derivatives with respect to the only independent variable each of these functions have, the chain rule is written as
$$ (g\circ f)'(x) = g'(f(x))\cdot f'(x) = (g'\circ f)(x)\cdot f'(x).$$
A: In short: Yes you may, but if you have more variables it becomes a mess. You should name the variable differently to emphasize that these come from a different set:
$$g \circ f : \mathbb R \ni x \stackrel{f(x)}\rightarrow \mathbb R \ni y \stackrel{g(y)}\rightarrow \mathbb R$$
Thus, for practical reasons, I recommend writing
$$\def\d{{\rm d}} \frac\d{\d x} g(f(x)) = \frac\d{\d y} g(f(x)) \cdot \frac\d{\d x}f(x)$$
This also saves on the parentheses, since the variable $y$ means to differentiate $g$ and evaluate at $f(x)$. A more complete notation would be
$$\left.\frac\d{\d t}(g\circ f(t))\right|_{t=x} = \left.\frac\d{\d s} g(s)\right|_{s=f(x)} \cdot \left.\frac\d{\d t} f(t)\right|_{t=x}$$
A: $a$ is a function of $b$ means that a depends on b. If now b depends on c then, obviously c affects b and, thus, ultimately a depends on c. I think that y=gf is called a composition. In mathematics, function composition is the pointwise application of one function to another to produce a third function. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x. 
